Onboarding restore use core api

[wendevlin]

Proposed change

• Use new core onboarding api to upload and restore backups

  • reimplementation of restore process
  • reuse components from backup dashboard

• core PR: Add backup endpoints to the onboarding integration core#136051

• core PRs persist restore info

  • Persist backup restore status after core restart core#136838
  • Persist hassio backup restore status after core restart core#136857

Type of change

• Dependency upgrade
• Bugfix (non-breaking change which fixes an issue)
• New feature (thank you!)
• Breaking change (fix/feature causing existing functionality to break)
• Code quality improvements to existing code or addition of tests

Example configuration

Additional information

• This PR fixes or closes issue: fixes #
• This PR is related to issue or discussion:
• Link to documentation pull request:

Checklist

• The code change is tested and works locally.
• There is no commented out code in this PR.
• Tests have been added to verify that the new code works.

If user exposed functionality or configuration variables are added/changed:

• Documentation added/updated for www.home-assistant.io